FIBONACCI NUMBERS CONSEQUENCES OF FIBONACCI NUMBERS HOW MANY

FIBONACCI NUMBERS CONSEQUENCES OF FIBONACCI NUMBERS

HOW MANY OF YOU KNOW ABOUT FIBONACCI NUMBERS?

FIBONACCI SEQUENCE 0

FIBONACCI SEQUENCE 0, 1

FIBONACCI SEQUENCE 0, 1, 1

FIBONACCI SEQUENCE 0, 1, 1, 2

FIBONACCI SEQUENCE 0, 1, 1, 2, 3

FIBONACCI SEQUENCE 0, 1, 1, 2, 3, 5

FIBONACCI SEQUENCE 0, 1, 1, 2, 3, 5, 8

FIBONACCI SEQUENCE 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ………

FIBONACCI SEQUENCE 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ……… Fn =Fn 1+ Fn 2 where n>2

HISTORY OF FIBONACCI SEQUENCE • The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. • This sequence was initially introduced in India by Indian ancient mathematicians around 1300 years back. • This sequence was introduced to west in 1202 by Leonardo of Pisa AKA Fibonacci.

ORIGIN OF FIBONACCI NUMBERS IN ANCIENT INDIA • Fibonacci Numbers were used in SANSKRIT POETRY. • Fibonacci Numbers were called "Matrameru" in Ancient Medieval India. • Fibonacci numbers were first used by ‘PINGALA’ in his Chandaḥśāstra which is the earliest known treatise on Sanskrit poetry.

SYLLABLES • Syllables(akṣara) in Sanskrit are generally represented by a single sound created by the pronunciation of vowels. There are 13 vowels in Sanskrit alphabets and hence 13 syllables. • A syllable must have a vowel, and it must have only one vowel. • They are categorized into two groups short or light (laghu) and long or heavy (guru). • There are five laghu syllables and eight guru syllables. • The guru (long) syllables takes exactly twice as long to say laghu (short) syllables. • So, we can define time as a unit and therefore guru syllables takes 2 units and laghu syllable takes 1 unit. • These units in Sanskrit are termed as Mātrā (Mora).

SANSKRIT POETRY • In Sanskrit poetry, we have stanzas with four quarters. Each quarter is assigned certain number of time units AKA Mātrā. • Now, suppose you are given one unit or Mātrā.

Then in how many ways you can arrange the laghu and guru syllables, remember that laghu takes one unit of time and guru takes two units of time. laghu 1 way

For two Mātrā. laghu and guru Two ways For simplicity let us denote laghu by ‘L’ and guru by ‘G’ Therefore for one Mātrā L 1 way for two Mātrā L L and G 2 ways for three Mātrā L L L , L G and G L 3 ways

Similarly for 4 Mātrā L L , L L G , G L L , L G L an d 5 ways G G for 5 Mātrā L L L , G G L L G , G L L L , G L G and L G G , L L G L , L G L L , 8 ways for 6 Mātrā LLLLLL, LLLLG, GLLLL, LGLLL, LLGLL, LLLGL, GGLL, LLGG, GLLG, GGG, LGLG, GLGL and LGGL – 13 ways for 7 Mātrā 21 ways for 8 Mātrā 34 ways for 9 Mātrā 55 ways for 10 Mātrā 89 ways ……………. .

for 1 Mātrā 1 way for 2 Mātrā 2 ways for 3 Mātrā 3 ways for 4 Mātrā 5 ways for 5 Mātrā 8 ways for 6 Mātrā 13 ways for 7 Mātrā 21 ways for 8 Mātrā 34 ways for 9 Mātrā 55 ways for 10 Mātrā 89 ways for 11 Mātrā 144 ways for 12 Mātrā 233 ways and so on ……………. . This is what ? Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ……… Whom do we appreciate? Fibonacci or PINGALA

Now let us play with Fibonacci numbers Let us square some Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ……. 1, 1, 4, 9, 25, 64, 169, 441, 1156, 3025, 7921, ………. We know that, as we add the two consecutive terms of Fibonacci sequence we get the next term of the sequence. Now we perform that simple trick on the sequence of the square of the Fibonacci numbers…. 1+1=2 4+1=5 9+4=13 25+9=34 64+25=89 And so on …………………. What is this? These are the terms of Fibonacci sequence with the gap of one term in between.

Another one……. . Suppose we add the squares of first few terms of Fibonacci sequence…. . 1+1+4=6 =2*3 1+1+4+9=15 =3*5 1+1+4+9+25=40 =5*8 1+1+4+9+25+64=104=8*13 ………………. 1+1+4+9+25+64+169=273=13*21 Nothing interesting…… Okay…. . Let’s see. There are Fibonacci numbers……. . But why this is true …. Let’s see.

21 8*8 13 13*13 2*2 1*1 5*5 3*3 That’s why the sum of the squares of the first few terms is the product of Fibonacci numbers

Okay now let’s see, Where do we find Fibonacci numbers in nature? In pine cones: There are spirals in both directions – clockwise and counter clockwise. As you can see in the clockwise direction there are 13 spirals and in counter clockwise direction there are 8 spirals. This is not the accident, actually, you will always find that, the number of spirals in both the direction are adjacent Fibonacci numbers

In sunflower seed head: In sunflower seed head also, the number of spirals in both direction are the adjacent Fibonacci numbers.

Inside the fruit of many plants we can observe the presence of Fibonacci order. You can see that the number of inner patterns are Fibonacci numbers.

Fibonacci in flowers: If you count the number of petals in any flower, not always, but most of the time you will find that the number of petals is a Fibonacci number.

SPIRALS Till now we have seen many spirals, but how Fibonacci numbers leads to spirals Okay… let’s see. 21 8*8 13 13*13 2*2 1*1 5*5 3*3 This is not accurate, but this is the main idea that how Fibonacci numbers leads to spirals.

GOLDEN RATIO Okay…. Now we should play another trick, divide the (n+1)th term of Fibonacci sequence by nth term of the Fibonacci sequence. Let’s do it…. 2 /1=2 3 /2=1. 5 5 /3=1. 66666667 8 /5=1. 60000000 13 /8=1. 6250000000 21 /13=1. 61538462 34 /21=1. 61904762 55/34=1. 61764705882353 89/55=1. 6181818 2 144/89=1. 61797752808989 233 /144=1. 6180555556 And so on……… This converges to the GOLDEN RATIO. phi =1. 618033988749895. . .

GOLDEN RATIO The GOLDEN RATIO is defined as: Inverse of the length of the larger part of the line, such that the ratio of length of the line to the length of larger pa The ratio of the length of larger part to the smaller part. x 1 x Let a line of unit length and the length of larger part is ‘x’ then the length of the smaller part is ‘ 1 x’. According to the definition : 1/x = x/1 x

GOLDEN RATIO Can anyone solve this two series…. . (Hint : these to series are equal to the golden ratio)

FIBONACCI NUMBERS AND FRACTALS We are only familiar with the geometry in integral dimensions. For example, A line, one dimensional geometry. A circle, two dimensional geometry. A cube, three dimensional geometry. • But there is a world beyond that, (any guesses, what it can be) • That world is of FRACTAL GEOMETRY, where the dimensions are in fractions. • FRACTAL GEOMETRY is the very new discipline of geometry which deals with the structure of the objects which we encounter daily. • For example, leaf of a plant, a whole tree, neurons in brain, mountains, stones, snow flakes etc.

HOW ARE FRACTALS PRODUCED? • To create a fractal, you can start with a simple pattern and repeat it at smaller scales, again and again, forever. In real life, of course, it is impossible to draw fractals with “infinitely small” patterns. However we can draw shapes which look just like fractals. Using mathematics, we can think about the properties a real fractal would have – and these are very surprising. Simply a defined iteration…….

FRACTALS IN INDIAN TEMPLES

MANDELBROT SET Why MANDELBROT SET? We need to introduce complex numbers for this (why? ) Mandelbrot set is a set of complex numbers ‘c’ such that when iterating the complex equation with 0 the Modulus of the result after each iteration is always less than or equal to 2.

PROBLEM: Where do Fibonacci sequence occurs in Mandelbrot set and why? But we can discuss.

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